T1 and T2 of protein and water protons in [human] Hb solutions were measured using broad-line pulse technique: selective excitation and detection methods enabled the intrinsic protein and water relaxation rates, as well as the spin-transfer rate between them, to be obtained at 5, 10 and 20 MHz. Water and protein T1 data were also obtained at 100 and 200 MHz for Hb in H2O/D2O mixtures by using commercial Fourier-transform instruments. The T1 data conform to a simple model of 2 well-mixed spin systems with singel intrinsic relaxation times and an average spin-transfer rate, with each phase recovering from a radio-frequency excitation with a biexponential time dependence. At low frequencies, protein T1 and T2 agree reasonably with a model of dipolar relaxation of an array of fixed protons tumbling in solution, explicitly calculating methyl and methylene relaxation and using a continuum approximation for the others. Differing values in H2O an D2O are mainly ascribed to solvent viscosity. For water-proton relaxation, T1, T2 and spin transfer were measured for H2O and HDO, which enabled a separation of inter and intramolecular contributions to relaxation. Despite such detail, few firm conclusions could be reached about hydration water. But it seems clear that few long-lived hydration sites are needed to explain T1 and T2, and the spin-transfer value mandate < 5 sites with a lifetime > 10-8 s.